Ukrainian Mathematical Bulletin

Volume 3, Issue 1, 2006 pp. 31-44.

Growth and Representation of Analytic and Harmonic Functions in the Unit Disc

Authors: Ihor Chyzhykov
Author institution: Department of Mechanics and Mathematics, Lviv Ivan Franko National University, Universytets'ka 1, 79000, Lviv, Ukraine, ichyzh@lviv.farlep.net

Summary: Let $u(z)$ be harmonic in $\{ |z| 1\}$, $\al \ge 0$, $0 \ga\le 1$. Let $B(r,u)=\max\{ u(z): |z|\le r\}$, $\om(\de, \psi)$ be the modulus of continuity of a function $\psi$ defined on $[0,2\pi]$. We prove that $u(z)$ has the form \begin{equation*} u(re^{i\vfi})=\frac 1{2\pi} \int\limits_0^{2\pi} P_\alpha(r, \vfi-t)\, d\psi(t), \end{equation*} where $\psi \in BV[0,2\pi]$ and $\om(\de,\psi)=O(\de^\ga)$, $\de\downarrow 0$, if and only if $ B(r, u )=O((1-r)^{\gamma -\al -1}), \ r\uparrow 1 $, and $ \sup _{0 r 1} \int _{0}^{2\pi} |u_\alpha(re^{i\vfi})|\,d\vfi +\infty$. Here $u_\al$ is the $\al$-fractional integral of $u$, $P_{\al}(r,t)=\Gamma(1+\al) \Re ( \frac 2{(1-re^{it})^{\al+1}} -1)$.

Contents Full-Text PDF